Symplectic Lefschetz Fibrations and the Geography Of

Symplectic Lefschetz ﬁbrations and the geography of symplectic 4-manifolds

Matt Harvey

January 17, 2003

This paper is a survey of results which have brought techniques from the theory of complex surfaces to bear on symplectic 4-manifolds. Lefschetz fibrations are defined and some basic examples from complex surfaces discussed. Two results on the relationship between admitting a symplectic structure and admitting a Lefschetz fibration are explained. We also review the question of geography: 2 which invariants (c1, c2) occur for symplectic 4-manifolds. Constructions of simply connected symplectic manifolds realizing certain of these pairs are given.

1 Introduction

It is possible to answer questions about a broad class of 4-manifolds by using the fact that they admit a special kind of structure called a Lefschetz fibration: a map from the manifold to a complex curve whose fibers are Riemann surfaces, some of them singular. The interesting data in such a fibration is the number of singular fibers and the types of these singular fibers. The types can be specified by monodromies (elements of the mapping class group of a regular fiber) associated to loops around the critical values of the fibration map in the base curve.

Lefschetz fibrations should, at least in theory, frequently be applicable to questions about symplectic 4-manifolds, in light of a theorem of Gompf that a Lef- schetz fibration on a 4-manifold X whose fibers are nonzero in homology gives a symplectic structure on X and a theorem of Donaldson that, after blowing up a finite number of points, every symplectic 4-manifold admits a Lefschetz fibration.

After giving some examples from the category of complex surfaces and discussing the above results on Lefschetz ﬁbrations, we state and sketch a proof of Gompf’s theorem that ﬁber sums can be carried out along codimension 2 sym-

1 plectic submanifolds so that the resulting manifold will still carry a symplectic structure.

Next is a brief introduction to the geography problem. The interesting part of this problem in the symplectic category is the question of existence of symplectic manifolds which are not homeomorphic to any complex surface. We will see that such manifolds do exist, and in fact all possible invariants of spin symplectic manifolds lying below the so-called Noether line are easy to realize using Gompf’s symplectic ﬁber sum.

There are a few conventions used throughout the paper. The word curve always means complex curve (dimR = 2), but surface means closed 2-manifold. A complex surface will always be explicitly referred to as such. Coefficients for homology and cohomology are taken in Z when not otherwise specified. The Euler class e(ξ) of a principal U(1) = SO(2) bundle ξ is the same as the first Chern class. We may evaluate e(ξ)[S2] for such a bundle over the 2-sphere and speak of the Euler number. This can be viewed in terms of building total space of the associated disc bundle to the given principal bundle from two copies of D2 × D2. In particular, in the first factor of this product, the construction is that of S2 as the union of the “northern” and “southern” hemispheres along the “equator.” The fibers in the second factor are identified by assigning an element of SO(2) to each point of the equator. By a diffeomorphism, we can choose the element of SO(2) assigned to some fixed base point on the equator to be the identity. The identifications over the equator then give a map S1 → SO(2). The isomorphism type of the principal bundle depends only on the homotopy class ∼ of this map, which is an element of π1(SO(2)) = Z called the Euler number of the bundle. PD(·) is used in various contexts to denote the Poincarédulaity ∼ n−k isomorphism Hk(X) = H (X) for compact manifolds. We sometimes abuse notation and identify the latter group with the deRham cohomology.

2 Building blocks in the complex category

The following definition will be needed later, but it also provides one of the most basic examples of a 4-manifold admitting a Lefschetz fibration. (This particular one has no critical points and no singular fibers.)

Deﬁnition 2.1 A ruled surface is a compact complex surface S with a map 1 π : S → C to a complex curve C such that every ﬁber is a complex line CP .

2 2.1 Lefschetz ﬁbrations

The structure of Lefschetz ﬁbrations comes from a decomposition of complex surfaces called a Lefschetz pencil.

n If S ⊂ CP is a complex surface, then a generic codimension 2 linear subspace n A ⊂ CP meets S in some ﬁnite set of points B. We may then consider two n codimension 1 linear subspaces of CP which both contain the subspace A but otherwise are generic. We can specify these subspaces as the zero sets of two linear homogeneous polynomials V (p0) and V (p1).

1 Now for fixed t = [t0 : t1] ∈ CP , consider the variety Lt = V (t0p0 + t1p1). This n will be a codimension 1 linear subspace of CP containing A. By a dimension count, we see that Lt intersects our original surface S in a complex curve, 1 which may be singular. Choosing s, t ∈ CP , note that Ls ∩ Lt = A so that (Ls ∩ S) ∩ (Lt ∩ S) = B. That is, all of the curves obtained above intersect exactly in the finite set of points B. After blowing up each point of B, we obtain 2 1 1 a well-defined map X#|B|CP → CP by sending each point to that t ∈ CP specifying the curve Lt ∩ S on which it lies.

Deﬁnition 2.2 A Lefschetz ﬁbration on a 4-manifold X is a map π : X → Σ where Σ is a closed 2-manifold having the following properties:

1. The critical points of π are isolated.

2. If P ∈ X is a critical point of π then there are local coordinates (z1, z2) on X and z on Σ with P = (0, 0) and such that in these coordinates π is 2 2 given by the complex map z = π(z1, z2) = z1 + z2 .

In terms of this descrtiption of the critical points, a Lefschetz ﬁbration can be thought of as a kind of complex analogue of a Morse function.

Note that if the 4-manifold X is closed (∂X = ∅ and X is compact) then the generic fiber F of π is a closed 2-manifold, so a Riemann surface of some genus. Part of the utility of Lefschetz fibrations is that this simple property means that X can be reconstructued once we know how the fibers are identified if one travels around some loop in Σ \ C where C ⊂ Σ is the set of critical values of π. That is, Lefschetz fibrations admit a combinatorial description in terms of their monodromy representation ρ : π1(Σ \ C) → π0(Diff(F )) from the fundamental group of Σ \ C into the group of homotopy classes of self-diffeomorphisms of F . The latter group is often called the mapping class group.

3 2.2 Examples

2 2 A very simple example of a Lefschetz fibration is a map from CP #CP → 1 2 CP . One begins with a point P ∈ CP and two generic lines containing this point, which we can take to be the vanishing sets V (p0) and V (p1) of two 2 independent degree 1 polynomials. For any point Q ∈ CP \ P there is a unique line V (pQ) containing both P and Q given as the vanishing set of another degree 1 polynomial. Some easy linear algebra shows that one can write pQ = t0p0+t1p1 1 for some [t0 : t1] ∈ CP . The map Q 7→ [t0 : t1] gives the desired fibration on 2 CP \ P . Since we can view blowing up the point P as replacing the point P by the set of all lines passing through P , we obtain a well-defined fibration 2 2 1 2 2 CP #CP → CP . Note that this shows CP #CP is a ruled surface since the 1 fiber over a point [t0 : t1] is V (t0p0 + t1p1), a CP .

2 One can play a similar game starting with points P1,...,P9 ∈ CP given as the intersection V (p0)∩V (p1), where p0 and p1 are two generic homogeneous cubics in three variables. The nine intersection points are in general position. We will be interested in vanishing sets of polynomials of the form

3 3 3 2 2 a1X + a2Y + a3Z + a4X Y + a5X Z+ 2 2 2 2 a6Y Z + a7Z X + a8Z Y + a9Y X + a10XYZ.

2 Picking a point Q ∈ CP \{P1,...,P9} and asking that its coordinates [X : Y : Z] satisfy the above equation gives one linear relation on the ai. In the same way P1,...,P9 give 9 more. Generically, we obtain in this way a homogeneous cubic which (by more linear algebra) can be written as pQ = t0p0 + t1p1. Note 0 that all points Q ∈ V (pQ) give the same [t0 : t1]. Again, by viewing the blow-up of each of the Pi as replacing that point by the set of all points passing through 2 2 1 it, we obtain a map CP #9CP → CP whose generic fibers are elliptic curves 2 in CP , which are topologically the two-dimensional torus T 2. A holomorphic map π : S → C from a complex surface to a complex curve whose generic fibers are nonsingular elliptic curves is called an elliptic fibration. The above is an example known as E(1). There is also the notion of C∞ elliptic fibration which is roughly a map from a 4-manifold to a 2-manifold which is locally diffeomorphic to a holomorphic elliptic fibration by diffeomorphisms of open subsets of S and C commuting with the given maps S → C.

These two examples help one picture the more general construction. Given a holomorphic line bundle L → X over a complex manifold X with two sections σ0, σ1 ∈ Γ(X; L) which are transverse to the zero section, one can consider the 1 1 vanishing of t0σ0 + t1σ1 where [t0 : t1] ∈ CP to get a map X \ A → CP , where A = V (σ0) ∩ V (σ1).

4 2.3 Fiber sum

There is a way to piece together two Lefschetz fibrations π1 : X1 → Σ1 and π2 : X2 → Σ2 whose generic fibers have the same genus to obtain a new Lefschetz fibration π : X1#φX2 → Σ1#Σ2.#φ denotes the fiber sum that is about to be defined and # the ordinary connected sum of the base spaces. One begins with neighborhoods νi of generic fibers Fi of πi for i = 1, 2. These are diffeomorphic 2 to D ×Σg where Σg is the Riemann surface with the same genus as the Fi. One then picks a diffeomorphism φ : S1×Σg (orientation-reversing) of the boundaries of Xi \ νFi and identifies them via φ.

A special case that should be mentioned here is E(2) = E(1)#φE(1) with φ : S1 ×T 2 → S1 ×T 2 the identity map. Topologically this is a K3 (or Kummer) 3 surface, homeomorphic to a degree 4 hypersurface in CP . More generally we may inductively form the elliptic surface E(n) = E(n − 1)#φE(1) with φ the identity as in the construction of E(2). We will return to these examples in the discussion of geography.

3 Symplectic versions of these constructions

3.1 Lefschetz ﬁbrations on symplectic manifolds

Symplectic manifolds (X, ω) for which ω ∈ H2(X; R) is an integral class admit topological Lefschetz pencils via the construction for complex manifolds described in the previous section. That is, they are unions of the vanishing sets of ratios of two sections of a certain line bundle over X as the ratios vary 1 over CP . These real codimension 2 submanifolds all intersect exactly in the common zero set A of the two chosen sections. The set A has real codimension 4.

Theorem 3.1 (Donaldson) If (V, ω) is a symplectic manifold with [ω] integral then for large k, V admits a topological Lefschetz pencil, in which the ﬁbers are symplectic subvarieties representing the Poincar´edual of k[ω].

To obtain the structure of a Lefschetz ﬁbration, we will need to blow up X along this set A, which for a 4-manifold is just a ﬁnite set of points. Some remarks on the proof of the above theorem follow. First note that the requirement that ω be an integral class (which seems to show up as a hypothesis of many other results in addition to this one) is not so severe.

Proposition 3.1 [Gom95] Any closed symplectic manifold (M, ω) admits another symplectic form ω0 whose cohomology class is in the image of H2(M, Z) →

5 2 HDR(M).

Proof. Fix a metric on M and let B be the -ball about 0 in the space of harmonic1 2-forms on M. Since nondegeneracy is an open condition, all of the forms in ω + B will be symplectic when is sufficiently small. Since ω + B 2 2 covers an open subset of HDR(M) it contains an element of H (M; Q), which 0 we may multiply by a suitable integer to get the desired ω . Holomorphic sections were used to pick out the fibers in the construction of a Lefschetz fibration on a complex manifold, so all the fibers also have the structure of complex manifolds. One can pick a compatible almost complex structure J on any symplectic manifold (X, ω), but there is no guarantee that there will be any complex line bundle with a sufficient supply of J-holomorphic sections to construct the Lefschetz pencil. Donaldson remedies this defect by considering an appropriate line bundle with sections which are “approximately holomorphic.”

ω Since ω is an integral class, we can form a complex line bundle L with c1(L) = 2π and pick a unitary connection on L with curvature −iω. This lets one deﬁne a ∂¯ operator on the sections of L.

Theorem 3.2 [Don96] Let L → X be a complex line bundle over a compact symplectic manifold (X, ω) with compatible almost-complex structure, and with ω c1(L) = 2π . Then there is a constant C such that for all large k there is a section s of L⊗k with C |∂s¯ | < √ |∂s| k on the zero set of s. Where ∂ and ∂¯ are the complex linear and anti-linear parts of the derivative ∇ which is canonically deﬁned along the vanishing of any section of L.

We can apply the following proposition on each tangent space to the vanishing set of a section to see that for a section s ∈ Γ(X; L⊗k) satisfying |∂s¯ | < |∂s| we have s−1(0) ⊂ M a symplectic submanifold.

Proposition 3.2 Suppose that A : Cn → C is a real linear map. Let a0 and a00 be the complex linear and conjugate linear parts of A. Then if |a00| < |a0| then ker(A) is symplectic with respect to the standard symplectic structure ω(v, w) = Im hv, wi, where h·, ·i is the standard Hermitian metric on Cn. 1This means those η ∈ Ω2(M) such that 4η = (dd∗ + d∗d)(η) = 0 where d is exterior diﬀerentiation and d∗ is its adjoint with respect to the bilinear forms given on Ω∗(M) by the metric. We view d, d∗ :Ωeven(M) → Ωodd(M). It is a theorem that every deRham cohomology class has a unique harmonic representative.

6 3.2 Symplectic structure on Lefschetz ﬁbrations

The converse to the result in the last section is considerably easier to prove. The argument below follows [GS99].

Theorem 3.3 Suppose that X is a closed 4-manifold with a Lefschetz fibration π : X → Σ. Denote the homology class of a regular fiber by [F ] ∈ H2(X). Then X admits a symplectic structure which is symplectic on all the fibers of π if and only if [F ] 6= 0.

Proof. [F ] 6= 0 is necessary, since given a symplectic form ω ∈ Ω2(X) which is R also symplectic on ﬁbers we would have F ω > 0 for F a regular ﬁber.

2 R If [F ] 6= 0 then there exists some α ∈ Ω (X) with F α > 0. While this is a big step towards constructing a symplectic form on X which is also symplectic R on the fibers, we have to make sure that α > 0 for each closed surface F0 F0 contained in a singular fiber S, which is topologically a regular fiber with some loops called vanishing cycles collapsed to points.

Singular fibers are still homologous to regular fibers. If S = π−1(c) is the preimage of a critical value c ∈ Σ of π, and F = π−1(d) is a regular fiber, then considering the preimage of an arc from c to d in Σ shows that [S] = [F ] ∈ H2(X). The same argument shows that all the regular fibers are homologous.

If a singular fiber S is obtained from F by collapsing a separating vanishing R cycle, then we have S = F0 ∪ F1 for some closed surfaces F0 and F1 and F α = R α + R α > 0, so if R α = r ≤ 0 then s = R α > 0. We can replace α by F0 F1 F0 F1 0 s R R 0 s α = α+(−r+ )PD(F1). PD(F1) = [F0]·[F1] = 1, so α = r−r+ > 0 2 F0 F0 2 R 0 s and α = s + (−r + )[F1] · [F1] > 0. All other fibers F of π do not intersect F1 2 R 0 R F1 and so have F α = F α. Since there are only finitely many isolated critical points of π we obtain, after finitely many modifications of this kind, a closed 2 R form ζ ∈ Ω (M) such that ζ > 0 for each closed surface F0 contained in a F0 fiber of π.

Now we need to fix this to be symplectic on fibers. Pick disjoint open balls Uj ⊂ X about each critical point of π so that on each Uj there are local coordinates 2 2 with in which we have π(z1, z2) = z1 + z2 . We have the standard symplectic structure ωUj = dx1 ∧ dy1 + dx2 ∧ dy2 on each Uj. π is holomorphic, so if y ∈ Σ −1 2 then π (y) ∩ Uj is a holomorphic curve, which means it is ωUj -symplectic. −1 For fixed y, extend the symplectic structure on π (y) ∩ ωUj to a symplectic −1 −1 structure ωy on all of π (y). (Extend means pick an ωy if π (y) ∩ Uj = ∅.) R R We can rescale each ωy staying away from ∪Uj to make F ωy = F ζ so that 2 [ωy] = [ζ] ∈ HDR(X; R) for all y. Now we need to glue these together. 2In general if J is an ω-compatible almost complex structure then C is J-holomorphic if and only if ω|C is symplectic.

7 Pick a finite cover {Wj} of Σ with at most one critical value in each Wj. Pick yj ∈ Wj for each j, choosing yj to be the critical value for those Wj which −1 contain a critical value. If π (Wj) is just a collection of regular fibers, let −1 −1 −1 rj : π (Wj) → π (yj) be a retraction. If π (yj) is a singular fiber, then take −1 −1 3 a retraction rj : π (Wj) → π (y) ∪ cl(Uj). Choose also a partition of unity P 2 ρj = 1 subordinate to this cover. We can define forms ηj ∈ Ω (Wj) for each j by ∗ r ωUj (P ), r(P ) ∈ cl(Uj) ηj(P ) = ∗ r ωyj (P ), otherwise. 1 −1 Since we made the [ωy] = [ζ] there exist θj ∈ Ω (π (Wj)) so that

−1 (ωy − ζ) |π (Wj ) = dθj for all j. We can then form

X 2 η = ζ + d (ρj ◦ π)θj ∈ Ω (X). P This has dη = dζ = 0 and so is a closed form. For a given fiber ηFy = ρj(y)ηj is linear combination of forms sympelctic on Fy with coefficients in [0, 1] and so is itself symplectic on Fy. To get a symplectic form on X, let ∗ ωt = tη + π ωΣ, where ωΣ is any symplectic form you choose on the base. ωt is closed for all t since η and ωΣ are closed. Along fibers, ωt agrees with tη and so is nondegen- −1 erate. If x ∈ π (y) is not a critical point of π then for v, w ∈ TxFy we have ∗ ωt(v, w) = tη(v, w) since π ωΣ is zero along fibers. This also shows that we have ωt η equal symplectic orthogonals (TxFy) = (TxFy) ⊂ TxX. Since η is symplectic ωt ∗ on Fy we have TxX = TxFy ⊕ (TxFy) , and π ωΣ is nondegenerate on the second factor. Since nondegeneracy is an open condition, ωt is nondegenerate on TxX for small enough t.

The argument above worked away from the critical points of π. In particular, since X\∪Uj is comapct, there is some t0 with ωt symplectic for 0 < t ≤ t0. Now ∗ on the Uj we have (ωt)|Uj = tωUj +π ωΣ. Recall that on the Uj we have complex structures on base and total space such that π is a holomorphic map. We can choose ωΣ so that the almost complex structure on the base is compatible, and we have ωUj compatible with multiplication by i in Uj by deﬁnition of ωUj . If v ∈ TUj is given in the local coordinates (z1 = x1 + iy1, z2 = x2 + iy2) by v = a ∂ + b ∂ + c ∂ + d ∂ then ∂x1 ∂y1 ∂x2 ∂y2

∗ ωt(v, iv) = tωUj (v, iv) + π ωΣ(v, iv)

= t(dx1 ∧ dy1 + dx2 ∧ dy2)(v, iv) + ωΣ(π∗v, π∗(iv)) 2 2 2 2 = t(a + b + c + d ) + ωΣ(π∗v, iπ∗v) > 0 3cl(·) means the closure.

8 because it is the sum of two positive terms, so all the ωt are nondegenerate on ∪Uj as well as on its complement, so we have the desired symplectic form on X with all ﬁbers symplectic and see that it arises as a perturbation of the pullback ∗ π ωΣ of a symplectic form on the base.

3.3 Blowing up

Blowing up is a familiar procedure for removing non-transverse intersections and self-intersections of curves. One views this locally at P ∈ C2 by picking a neighborhood of P diﬀeomorphic to C2 in such a way that P = (0, 0). Then let

2 1 γ = {(Q, l) ∈ C × CP | Q ∈ l}.

1 2 Note that γ \ 0 × CP is diﬀeomorphic to C \{(0, 0)} via some map φ. E = 1 {0} × CP is called the exceptional sphere. We can extend φ smoothly by 0 deﬁning φE = 0. If P ∈ X is a point in a 4-manifold, we can form X = 0 (X \ P )∪φ γ which comes with an obvious map σ : X → X with E the preimage of P .

Proposition 3.3 The topological eﬀect of the blowup process is to replace the 2 4-manifold X with X#CP .

Proof. γ is clearly a complex line bundle over E via projection onto the second factor. Forming the blowup consists of removing a 4-ball from X and gluing in the unit disc bundle in γ over E = S2. The Euler number of this disc bundle is −1. To see this, note that the function (z1, z2) 7→ z1 is linear on every line through the origin in C2 and so can be thought of as a section of γ∗ = Hom(γ, C). It intersects the zero section of γ∗ exactly in [0 : 1] and is a holomorphic subset, so this intersection is positive. Now for general complex line bundles γ we have e(γ∗) = −e(γ), so we see the Euler number of our particular γ is −1.

2 2 The Euler number of the normal bundle of a hyperplane in CP is +1, so CP is obtained by adding a 4-ball to the boundary of a disc bundle with Euler 2 number −1 along it’s S3 boundary, so the connect sum with CP and the blowup process are both replacing a B4 with a D2 bundle over S2 with Euler number −1.

2 Proposition 3.4 If X is a symplectic 4-manifold then the blowup X#CP is also symplectic.

The following notion is useful in discussing geography problems:

9 Deﬁnition 3.1 A 4-manifold X is called minimal if there is no 4-manifold Y 2 with X = Y #CP , that is, if X is not the blowup of another manifold.

By work of Taubes, being nonminimal is equivalent to containing a smoothly embedded S2 with self-intersection −1. (The deﬁnition above requires this S2 to be symplectic.) One can also speak of a kind of minimality for Lefschetz ﬁbrations.

Definition 3.2 A Lefschetz fibration is called relatively minimal if there is no S2 with self-intersection −1 contained in a fiber.

Proposition 3.5 [Sti00] If π : X → Σ is a Lefschetz ﬁbration with g(Σ) ≥ 1 then π is a relatively minimal Lefschetz ﬁbration if and only if X is minimal.

Proof. Assume the result of Taubes that if X contains a sphere of −1 self- intersection there is a compatible almost complex structure on X for which there is a J-holomorphic (−1)-sphere. When X is equipped with such an almost- complex structure, π becomes a holomorphic map from the sphere to Σ and is therefore constant, so this (−1)-sphere is actually contained in one of the ﬁbers of π. This shows relative minimality of π implies minimality of X. The converse is obvious.

3.4 Gompf’s symplectic ﬁber sum

The main theorem of [Gom95] states that ﬁber sums can be carried out along symplectic submanifolds and the result will still carry a symplectic structure.

n n−2 Theorem 3.4 Let (M , ωM ) and (N , ωN ) be closed symplectic manifolds and let j1, j2 : N → M be symplectic embeddings with disjoint images. Sup- pose also that if ν1 and ν2 denote the normal bundles of j1(N) and j2(N) in M, respectively and e denotes the Euler class then e(ν2) = −e(ν1). Then, for ∼ any choice of orientation-reversing Ψ: ν1 = ν2, the manifold #ΨM admits a symplectic structure ω, which is induced by ωM after a perturbation near j2(N).

For an even more precise statement and the details of carrying out ﬁber sums on pairs, see [Gom95]. A synopsis of the proof given there for the basic result above comprises the rest of this section.

We begin by setting up a model for the gluing of tubular neighborhoods of the codimension 2 submanifolds j1(N) and j2(N). The model represents the local situation after the gluing and consists of an S2 bundle S over N so that N sits

10 inside S as zero section and as ∞ section. There is an SO(2) action on S fixing these copies of N which just rotates all the fibers around their axes. To get 2 on more precise footing, let E be an R bundle isomorphic to ν1. Then E is isomorphic to ν2 since ν1 and ν2 are identified by the orientation-reversing map Ψ.

There is now the crucial use of the map ι : D2 \ 0 → D2 \ 0 given by s 1 ι(x) = x − 1. π|x|2 which antisymplectically turns the punctured 2-disc inside out. If (ρ, φ) = q q 1 1 2 ι(r, θ) = r πr2 − 1, θ = π − r , θ then

∂ρ ρ dρ dφ = ρ dr dθ ∂r −r = ρ dr dθ ρ = −r dr dθ

Letting D+ be the sub disc bundle of E above consisting of the union of all the √1 closed normal discs of radius π and D− the disc bundle of discs with radius √1 π in E. We form the sphere bundle S by identifying D+ with D− via the orientation-reversing map ι on each disc. The SO(2) action mentioned earlier comes from the standard action on E and E. It is clear that this commutes with ι and so gives a well-deﬁned action on S. In the following N is sometimes referred to as a subset of D+ or D−. If i0 and i∞ denote the inclusions of N in S as the zero section and the ∞ section respectively, then N ⊂ D+ really means i0(N) ⊂ D+ = S \ i∞(N), and N ⊂ D− means i∞(N) ⊂ D− = S \ i0(N). It is hoped that this abbreviated notation will alleviate rather than cause confusion.

The construction makes use of a symplectic form η ∈ Ω2(S) with the following properties:

1. η is SO(2)-invariant. R 2. F η = 1 for any ﬁber F of S. ∗ 3. If i0 : N → S denotes inclusion as the zero section, then i0η = 0.

It is proved in [Gom95] that such an η exists and that if π : S → N is the projection then for some t1 the forms

∗ ωt = π ωN + tη

11 are SO(2)-invariant symplectic forms for t ∈ (0, t1]. With this setup, consider an embedding f : D+ → M with f ◦ i0 = j1 mapping the copy of N in D+ to our copy j1(N) in M. Such an embedding is guaranteed because we chose D+ to have the same Euler number as the normal bundle of j1(N) in M. Since j1 was assumed to be a symplectic embedding of N, f|i0(N) is symplectic. We can ∗ apply the Moser theorem to the forms ωE +t(f ωM −ωE) to get an isotopy from ˜ f to an embedding f which is symplectic on a neighborhood of N ⊂ D+. This isn’t good enough, though. What we really want is a symplectic embedding of the whole of D+. The following shrinking technique can be used to do this.

Proposition 3.6 For any neighborhood W of N in (D+, ωt1 ), there is a t0 with 0 < t0 ≤ t1 so that for all positive t ≤ t0, (D+, ωt) embeds symplectically in W ﬁxing N by an embedding isotopic (ﬁxing N) to the identity on D+.

Proof. The argument makes use of a standard chain homotopy operator, described in Section 6.3 of [CdS01]. We now recall that construction: For s ∈ [0, 1], let πs : D+ → D+ be given by multiplication by s on each fiber and de- d fine a vector field Xs = ds πs. One can then define an integral operator p p−1 I :Ω (D+) → Ω (D+) by Z 1 ∗ I(η) = πs (ιXs η) ds. 0

Using the nondegeneracy of ωt gives a time-dependent vector field Yt by ιYt ωt = 2 2 −I(η), where η ∈ ω (D+) is the restriction of the η ∈ ω (S) referred to above. We can then integrate Yt to an SO(2)-invariant flow on compact subsets of D+ that is defined for some open time interval about any fixed t0. The next step is to show that this flow is defined on all of D+ × [t0, t1].

For fixed x ∈ D+ denote by D(x) the disc in the fiber of D+ containing x which R is bounded by the SO(2) orbit of x. Define the area A(x) = D(x) η. Then by R R the definition of ωt above the ωt area of D(x) is D(x) ωt = t D(x) η = tA(x) ∗ since π ωN is zero along the fibers of D+. If Ft denotes the flow of the time- ∗ dependent vector field Yt defined above then Ft ωt is independent of t and the ωt area of Ft(D(x)) = D(Ft(x)) is given by Z Z Z tA(F (x)) = ω = F ∗ω = F ∗ ω = t A(x) t t t t t0 t0 0 Ft(D(x)) D(x) D(x) and we see that t A(F (x)) = 0 A(x). t t

In words, the disc fibers of D+ decrease in area under the flow Ft, so points cannot escape from D+ under the flow, and we get a well-defined Ft : D+ → D+ for t ∈ [t0, t1]. (We can choose 0 < t0 ≤ t1 as we please and Ft0 = IdD+ .) So

12 ¡ N j2(N) f(S−U) U f g

sphere bundle S

phi

phi g(U−N) f(U−N)

Figure 1: Fiberwise schematic for symplectic sum

we choose t0 suﬃciently small that Ft1 (D+) ⊂ W . This Ft1 is the desired symplectic embedding of the whole of D+ in W . ˜ Applying this to the embedding f : D+ → M which was symplectic on a neigh- ˆ borhood of N ⊂ D+ we get an embedding f : D+ → M onto a neighborhood of j1(N) that is symplectic on all of D+. We would like to do a similar thing for D− but must deal with the problem that i∞(N) ⊂ S is not necessarily symplectic with respect to ωt.

Let g : S \ i0(N) → M be the identification of the “northern” part of the sphere −1 bundle with a neighborhood of j2(N) in M. It is possible to extend g to a map λ from a neighborhood of the closure C of g(S \ i0(N)) in M sending all points ∗ outside of C to i0(N). We can form ζ = λ η to get a form on this neighborhood that extends by zero to all of M. We modify ωM toω ˜M = ωM + tζ. This is the perturbation referred to in the statement of the theorem. For small enough t,ω ˜t is symplectic on M and j2(N), and g|i∞(N) becomes a symplectic embedding. We can isotop g to be symplectic on a neighborhood U of i∞(N). Assuming that the embedding f : D+ → M is already symplectic on all of D+, which we just showed can be achieved by an isotopy of f, we are in the situation depicted in Figure 1. That is, f and g symplectically identify U \ N = U \ i∞(N) ⊂ S with g(U \ N) and f(U \ N). This latter is the same as f(D+) \ f(D+ \ U), so we finally see that we can modify M by removing a compact neighborhood f(D+ \ U) of j1(N) from M together with the copy j2(N) of N and gluing the “ends” (which look like open annulus bundles over N) together by the map φ = g ◦ f −1 which we have just seen is symplectic and which turns each annulus inside out as ι did to the punctured disc. Up to diffeomorphism, the effect is the same as the fiber sum described in the last section, but now it is furnished with a symplectic formω ˜t for some small t.

13 3.5 Non-K¨ahler manifolds

Some basic facts about K¨ahler manifolds are discussed in [CdS01]. We recall the deﬁnition given there:

Deﬁnition 3.3 A K¨ahler manifold is a symplectic manifold (M, ω) equipped with a compatible almost-complex structure J which is intergable.

Recall also that complex submanifolds of Kähler manifolds are Kähler and that n there is a Kählerform on CP , so that all nonsingular projective varieties are Kähler.

Proposition 3.7 The odd Betti numbers of a compact K¨ahler manifold are even.

Proof. It’s easy if one takes for granted that

k M l,m HDR(M; C) ' H (M) l+m=k Hl,m ' Hm,l where the group on the left comes from tensoring the ordinary deRham cohomology with C and the groups on the right are the Dolbeault cohomology 4 l,m l.m groups. If we let h = dimC(H (M)) then

2k+1 X i,2k+1−i b2k+1(M) = h i=0 k X = hi,2k+1−i + h2k+1−i,i i=0 k X i,2k+1−i = 2 h . i=0

In a short note [Thu76], Thurston gives the following example of a symplectic manifold with b1 odd. Consider the representation ρ : Z ⊕ Z → SL2Z given by 1 0 1 1 ρ(1, 0) = ρ(0, 1) = . 0 1 0 1

4Without the Kähler hypothesis there is still the Fröhlicher spectral sequence relating these 1 p,q groups with the page Ep,q = H (M). The fact used above is that this spectral sequence has E1 =∼ E∞ for a compact Kähler manifold. For more, see [GH94], which also states that there 2 ∞ are no known complex manifolds with E E .

14 ∼ 2 2 ∼ Note that Z ⊕ Z = π1(T ) and the mapping class group π0(Diff(T )) = SL2Z. Viewing this ρ as a holonomy representation gives a manifold X4 which is a T 2 bundle over T 2. We can view this as being built from a trivial T 2 bundle over S1 × I by identifying the fibers over the ends S1 × {0} and S1 × {1} via a diffeomorphism isotopic to the one specified by ρ(0, 1). We then have the following presentation for the fundamental group, which abelianizes to give H1(X) and so also the first Betti number b1(X).

4 ¯ ¯ π1(X ) = a, b, c, t | aba¯b = 1, aca¯c¯ = 1, bcbc¯ = 1, tat¯= a, tbt¯= ab a ⊕ b ⊕ c ⊕ t H (X) = Z Z Z Z 1 b = a + b 3 =∼ Z b1(X) = 3

By the above proposition, the manifold so constructed cannot be Kähler. It is a symplectic manifold by the result of Section 3.2, since a T 2 bundle over T 2 is a Lefschetz fibration with no singular fibers.

4 Geography

Since one can realize any finitely presented group as the fundamental group of some closed 4-manifold, a sub-problem of the classification of closed 4-manifolds in general is deciding if any given finite presentation of a group is in fact de- scribing the trivial group. This is known to be hard, so it is better to first ask for a classification of the simply-connected closed 4-manifolds.

In the topological category, this amounts to the classiﬁcation of intersection forms, thanks to the following:

Theorem 4.1 (Freedman) If Q is a unimodular symmetric bilinear form then there is some simply connected, closed 4-manifold X such that the intersection 5 form QX = Q. If Q is even then X is unique. Otherwise, there are two homeomorphism types of closed simply connected 4-manifolds having intersection form Q. At most one of these can have a smooth structure.

Note that for a closed 4-manifold X the intersection form must be unimodular, 2 ∗ H2(X) H (X) for if A = H2(X) then QX : A → A = Hom Torsion , Z = Torsion is an isomorphism by Poincar´eduality.

5 An integral form Q is called even if Q(x, x) ∈ 2Z for all x.

15 Theorem 4.2 [GS99] Indefinite intersection forms are classified by their rank, 0 1 signature, and parity and are equivalent to kH ⊕ lE where H = 8 1 0 and E8 is the Cartan matrix for the root lattice of the exceptional Lie algebra e8. If a definite intersection form belongs to a smooth, simply connected, closed 4-manifold then it is equivalent to ⊕k h±1i.

So the homeomorphism type of a smooth, simply connected, closed 4-manifold is determined by a few characteristic numbers, which will be discussed further in the next section. The simple connectivity hypothesis is important. T 2 × T 2 2 2 2 2 2 ∼ 4 ∼ 2 ∼ and T × S have the same (c1, c2), but π1(T × T ) = Z =6 Z = π1(Y ).

2 A geography problem asks which pairs of characteristic numbers (c1, c2) (or equivalently (χ, σ), as will be explained soon) actually occur as the invariants of a certain class of manifolds: which are realized by simply connected complex surfaces, which are realized by symplectic manifolds, etc.

4.1 Characteristic numbers

Any almost complex manifold comes with an almost complex tangent bundle, allowing one to deﬁne the Chern classes of the manifold. Recall that they are deﬁned inductively, with the top Chern class equal to the Euler class. The Chern class cn−1 is the top Chern class of the canonical complex (n − 1)-plane bundle one can construct over the complement of the zero section in a complex n-plane bundle.

For an almost complex (hence oriented) 4-manifold X, these characteristic classes yield two important characteristic numbers

2 c1 = h(c1(TX) ^ c1(TX)), [X]i and c2 = hc2(TX), [X]i

4 where h·, ·i denotes the natural pairing between H4(X) and H (X). Similarly, 4 the ﬁrst Pontryagin class p1(X) ∈ H (X) gives a characteristic number p1.

Note first that by definition c2 = χ(X) = χ is just the Euler characteristic of X. Here we will assume the Hirzebruch signature theorem, which states that p1 = 3σ for a smooth, closed, oriented 4-manifold. For any real vector bundle i ξ, we have the definition pi(ξ) = (−1) c2i(ξ ⊗ C). If ξ already had an almost complex structure, then there would be a natural identification ξ ⊗ C =∼ ξ ⊕ ξ¯, where ξ¯ denotes the bundle obtained from ξ by complex conjugation on each fiber. So we have ¯ p1(ξ) = −c2(ξ ⊕ ξ) ¯ ¯ = −(c2(ξ) + c1(ξ)c1(ξ) + c2(ξ) 2 = −(2c2(ξ) − c1(ξ)

16 2 c1(ξ) = p1(ξ) + 2c2(ξ). Applying this and the signature formula when ξ is the tangent bundle of a smooth, closed 4-manifold admitting an almost complex structure, we have 2 c1 = 3σ + 2χ.

Now σ, χ are defined for all 4-manifolds, so instead of using the definition in 2 terms of characteristic classes one may just define c1 = 3σ + 2χ and c2 = χ.

To see to what extent these characteristic numbers really characterize X, note + − + − that χ = 2 − 2b1 + b2 + b2 and σ = b2 − b2 . Thus for simply connected 4- manifolds (b1 = 0) the characteristic numbers give us the rank and signature of the intersection form.

2 Now c1 − 2c2 = 3σ. If X is spin (Roughly, if there is a trivialization of TX over the 1-skeleton of a CW structure for X which extends over the 2-skeleton.) then 2 2 by Rohlin’s theorem 16|σ so that 48|c1−2c2, which shows that most pairs (c1, c2) are not the invariants of a spin homeomorphism type. For simply connected spin manifolds, the rank and signature completely determine the homeomorphism type. For non-spin manifolds there are two homeomorphism types but only one of these admits a smooth structure. This is using the equivalence of X having an even intersection form with the vanishing of the second Stiefel-Whitney class 2 w2(TX) ∈ H (X; Z/2Z), which measures the obstruction to putting a spin structure on X.

By way of example, we summarize below the values of these invariants for some familiar 4-manifolds

2 + − c1 c2 = χ σ b2 b2 b1 = b3 2 CP 9 3 1 1 0 0 S2 × S2 8 4 0 1 1 0 E(1) 0 12 -8 3 11 0 E(2) = K3 0 24 -16 3 19 0 E(n) 0 12n -8n 2n-1 10n-1 0 2 Σg × S 8 − 8g 4 − 4g 0 1 1 2g T 2 × T 2 0 0 0 3 3 4

The usual way to pose a geography question for manifolds in a given category 2 is to ask which (c1, c2) can be realized by minimal manifolds in that category. This is explained by the following:

Proposition 4.1 The invariants of the blowup of a 4-manifold X are determined by the invariants of X.

2 2 2 2 c2(X#CP ) = c2(X) + 1 c1(X#CP ) = c1(X) − 1.

17 Proof. The ﬁrst statement follows from the fact that c2 = χ(X) is the Euler characteristic and that for any connected sum of manifolds M and N we have χ(M#N) = χ(M) + χ(N) − 2.

Using the fact that the signature of the intersection form adds under connect 2 sum and the formula for c1 above, we have the easy calculation

2 2 2 2 c1(X#CP ) = 3σ(X#CP ) + 2c2(X#CP ) = 3(σ(X) − 1) + 2(χ(X) + 1) 2 = c1(X) − 1.

2 In particular, the product Σ2 × Σk+1 of two Riemann surfaces has (c1, c2) = (8k, 4k), so by blowing up products of Riemann surfaces we can obtain all pairs 0 ≤ c1 ≤ 2c2 as the invariants of symplectic manifolds. However, these examples are not minimal, not spin, and not simply connected.

We just note here that there is a well-developed theory of the geography of simply connected minimal compact complex surfaces. In particular, these are projective varieties, so they are all symplectic. Thus the geography of simply connected minimal symplectic manifolds must contain that of the simply connected minimal complex surfaces. We will soon see that this is a proper containment.6 To put the constraints on the geography of symplectic manifolds in the following sections in perspective, we mention here constraints

2 c2 c1 ≥ 5 − 36 (the Noether line) and 2 c1 ≤ 3c2 (the Bogomolov − Miyaoka − Yau line) on minimal complex surfaces of general type. General type means that

dim Γ ((∧2T ∗X)⊗k) hol → ∞ k as k → ∞. (The Kodaira dimension is 2.) More details can be found in [BPVdV84].

4.2 Constraints on the geography of symplectic manifolds

2 One side of the geography problem is to identify regions of the (c1, c2)-plane where there is no hope of ever ﬁnding a (minimal) symplectic manifold. An important result in this direction is the so-called Noether condition, which uses only the almost-complex structure.

6Thurston’s example of a non-K¨ahler symplectic manifold given above is not enough to do this because that manifold was not simply connected.

18 Proposition 4.2 An almost-complex 4-manifold X has

2 c1 + c2 ≡ 0 (mod 12).

Proof. There is an argument relying on the classiﬁcation of integral unimodular 2 symmetric bilinear forms that c1 ≡ σ (mod 8). Details can be found in [GS99]. 2 2 This with c1 = 3σ + 2χ gives σ + χ ≡ 0 (mod 4) so that 3σ + 3χ = c1 + c2 ≡ 0 (mod 12).

One can also get this for complex surfaces via the Atiyah-Singer index theorem, 4 1 2 7 since the degree 2 term (the part in H (X)) of the Todd class is 12 c1 + c2 .

2 The following result rules out many of the possible pairs (c1, c2).

Theorem 4.3 [Liu96] Let (M, ω) be a symplectic 4-manifold which is not the 2 blowup of a ruled surface (see Deﬁnition 2.1). Then c1(M, ω) ≥ 0.

This is a consequence of the result of [Tau96] relating Seiberg-Witten invariants with Gromov invariants for symplectic 4-manifolds.

Given this lower bound, one might ask what kind of upper bound can be put on 2 c1. There is an easy result of this kind when b1 ≤ 1 (in particular it holds for simply connected manifolds) that does not require the presence of a symplectic structure.

2 Proposition 4.3 If X is a 4-manifold with b1(X) ≤ 1 then c1(X) ≤ 5c2(X).

2 Proof. Use c1 = 3σ + 2χ and c2 = χ as follows:

2 c1(X) = 3σ(X) + 2χ(X) + − = 3(b2 (X) − b2 (X)) + 2c2(X) − = 3(c2(X) − 2 + 2b1(X) − 2b2 (X)) + 2c2(X) − = 5c2(X) + 6(b1(X) − 1) − 2b2 (X) ≤ 5c2(X)

A basic question about the possible topology of symplectic 4-manifolds remains open:

7The author does not know whether this big machine is one that also works when the 1 2 almost-complex structure is not integrable. χh = 12 (c1 + c2), rather than c2, is sometimes used for the second characteristic number.

19 200

150

100

-100 100 200 300 400 500 600

-50

-100

Figure 2: Spin and simply connected symplectic manifolds

Conjecture 4.1 (Gompf) If X is a symplectic 4-manifold and X is not the blowup of a ruled surface, then the Euler characteristic χ(X) ≥ 0.

1 The qualiﬁcation is really necessary since the ruled surface Σg×CP is a symplec- 1 tic manifold with Euler characteristic χ Σg × CP = 4 − 4g. Given Theorem 4.3, this conjecture would also hold if the following one does, that is, if symplectic manifolds all fall below the BMY line as do the simply connected complex surfaces.

Conjecture 4.2 [Sti00] If X is a symplectic 4-manifold other than a sphere 2 bundle then c1(X) ≤ 3c2(X).

4.3 Constructions of symplectic 4-manifolds

A complete understanding of the geography of 4-manifolds carrying a symplectic structure requires not only constraints on the invariants of the type described 2 above but also proofs that pairs (c1, c2) satisfying the contraints actually occur as the invariants of symplectic manifolds. The most satisfying way to do this is to construct such manifolds explicitly.

20 After a short detour on slopes and arbitrary fundamental groups, we will review some constructions from [Gom95] that show that the invariants at the dots in the diagram are realized by spin symplectic 4-manifolds, and that all the 2 2 pairs (c1, c2) in the shaded region which satisfy the Noether condition c1 + c2 ≡ 0 (mod 12) are realized by simply connected symplectic manifolds. Not 2 all of these are minimal, but all the pairs in the shaded region with c1 ≥ 0 can be constructed without resorting to blowing up. Convention dictates that 2 we write pairs of invariants in the order (c1, c2) despite the fact that, also by 2 convention, the c1-axis is vertical and the c2-axis horizontal when depicting 2 geography graphically. The three lines in the diagram are the BMY line c1 = 2 2 1 3c2, the σ = 0 line c1 = 2c2, and the Noether line c1 = 5 c2−36. Some symplectic manifolds with positive signature are also discussed in [Gom95], but we will not discuss them here. Note, however, the abundant supply of spin symplectic manifolds lying below the Noether line. None of these are homeomorphic to minimal complex surfaces of general type.

Stipsicz notes in [Sti00] that a few of such constructions suﬃce to see that there are no gaps in the “slope” realizable by symplectic 4-manifolds.

Theorem 4.4 [Sti00] If α ∈ [0, 3] ∩ Q, then there is a minimal symplectic 4- 2 manifold X such that c1(X) = αc2(X). Consequently, there is no gap in the geography of symplectic 4-manifolds in the interval [0, 3].

The proof cites three constructions of symplectic 4-manifolds for each of the 1 1 three cases α ∈ [0, 5 ], α ∈ [ 5 , 2], and α ∈ [2, 3]. Gompf [Gom95] gives several constructions of symplectic manifolds as applications of his proof that the ﬁber sum operation can be carried out along codimension 2 symplectic submanifolds. One striking example of this is the following result on realizing a speciﬁed fundamental group:

Theorem 4.5 [Gom95] Let G be any ﬁnitely presented group. Then there is a closed, symplectic 4-manifold M with π1(M) = G.

Proof. (Sketch) As one might expect, M is constructed via a fancy version of the construction of a 2-dimensional CW complex having arbitrary ﬁnitely presented fundamental group. Instead of beginning with a wedge of circles, one begins with the product of a closed surface and a torus. If

G = hg1, . . . , gl | r1, . . . , rki then let F be the surface of genus l and {α1, . . . , αl, β1, . . . , βl} be embedded circles representing a symplectic basis for H1(F ). The relations r1, . . . , rk can be viewed as words in the αi alone, specifying circles γ1, . . . , γk in F .

21 Let S = E(n) \ ν(T 2) be one of the elliptic surfaces E(n) minus a neighborhood of a regular fiber. Then S is simply connected8, and a fiber sum with E(n) along 2 this regular fiber with a torus T in F ×T will have the effect of killing π1(T ) in the fundamental group. If µ is a meridian of the second factor of F × T 2 then we can get a closed 4-manifold X with π1(X) = G by fiber summing a copy of E(n) to each of the tori βi × µ for i = 1, . . . , l and γi × µ for i = 1, . . . , k and also summing a copy to a torus {pt} × T 2. This manifold will be symplectic if we can get a symplectic structure on F × T 2 for which all these tori are actually symplectic submanifolds.

2 ∗ ∗ If ω is the product symplectic structure on F × T and η = π1 ρ ∧ π2 α with α a volume form on µ, ρ a 1-form on F which restricts to a volume form on all the 2 2 2 βi and γi, and π1 : F × T → F and π2 : F × T → T the projections, then we can form ωt = ω + tη, and for suﬃciently small t this will be symplectic on the tori βi × µ and γi × µ. Before we are really in a place to do the ﬁber sum we have to perturb these tori to be disjointly embedded. It is possible, but the details are omitted here.

The other issue is whether the form ρ ∈ Ω1(F ) we asked for above actually exists. There are situations where it does not, but Gompf avoids this problem by taking copies of T 2 with distinguished curves a × S1, S1 × b, and S1 × c with b 6= c, designating a small disc D ⊂ T 2 intersecting c but not a ∪ b, and equipping these tori with 1-forms which are volume forms on each of the three distinguished curves and are zero near D. Connect summing one of these to each edge of the graph in F formed by the αi, βi, and γi in our original setup yields a new F with new αi, βi, and γi which still gives the same fundamental group. The point is that we may assume without loss of generality that the η we wanted in the last paragraph did exist. We now return to explaining the “map” above with the following:

Proposition 4.4 For a ﬁber sum M1#ψM2 along a surface of genus g we have

2 2 2 (c1, c2)(M1#ψM2) = (c1, c2)(M1) + (c1, c2)(M2) + (g − 1)(8, 4).

Proof. First note that we can view M1#ψM2 as having been obtained from (M1 × I) ∪ (M2 × I) by attaching handles to the top level and rounding corners. This gives a cobordism from M1 ∪ M2 to M1#ψM2, and since signature is a cobordism invariant we have

σ(M1#ψM2) = σ(M1) + σ(M2).

8It’s because we can contract the loop we added by cutting out the codimension 2 ﬁber along what remains of a section (S2 \{pt}) of the elliptic ﬁbration.

22 We have also χ(M1#ΨM2) = χ(M1) + χ(M2) − 2χ(N) where the sum is being taken along N. This gives the second component of the desired formula. The 2 ﬁrst follows easily from the formulas for σ and χ above together with c1 = 3σ + 2χ.

Theorem 4.6 [Gom95] Any pair of integers (m, n) satisfying the Noether and 2 Rohlin conditions and 0 ≤ m ≤ 2n is realized as (c1, c2) of a closed spin symplectic 4-manifold.

2 Proof. Recall that the Rohlin condition is 48|c1 −2c2. If 12|m+n and 48|m−2n then (m, n) = (8k, 4k + 24l) for some k, l ∈ Z. To prove the theorem, we need to realize such pairs with k, l ≥ 0. Taking the product of two Riemann surfaces of genus k + 1 and 2 with the product symplectic structure gives a symplectic 2 manifold with (c1, c2) = (8k, 4k). Picking a homologically nontrivial loop in each of the factors of this product gives a Lagrangian torus. We can take l parallel copies of such a torus so that they are disjointly embedded. It is possible to perturb the symplectic form so that these Lagrangian tori become symplectic, but we omit the argument here.

Carrying out the symplectic ﬁber sum with l copies of E(2) along its (torus) 2 ﬁber adds l(c1(E(2)), c2(E(2))) = (0, 24l) to our invariants, so we have seen how to realize all pairs (8k, 4k + 24l) as desired.

1 Theorem 4.7 [Gom95] If (m, n) are integers with n > 0 and −n + 16 2 ≤ m ≤ 2n − 33 then there is a simply connected symplectic manifold X with 2 (c1(X), c2(X)) = (m, n).

Proof. Like the last result, this is accomplished by ﬁber sums, but this time they are along surfaces of genus 2. We will need some building blocks.

2 Begin with a union of six lines in CP in general position. Resolve all but 8 points of intersection via deformation. Blowing up the 8 nodes will reduce the 2 2 self-intersection from 36 to 4. An additional 4 blowups yeild CP #12CP with a smoothly embedded surface of genus 2 having self-intersection 0. Call this manifold P . It is easily read oﬀ from the intersection form h1i ⊕ 12 h−1i that 2 c2(P ) = χ(P ) = 15 and σ(P ) = −11 so that c1(P ) = −3.

Consider the manifold Z described in Section 3.5. As we noted before, it is a T 2 bundle over T 2. It can be viewed as a quotient of R4 with the standard symplectic structure by the group generated by translations parallel to the x1, x2, and x3 axes and the (symplectic) map

(x1, x2, x3, x4) 7→ (x1 + x2, x2, x3, x4 + 1).

23 It fibers over T 2 via projection onto the last two factors. Moreover, it is par- allelizable, so χ(Z) = σ(Z) = 0. Let Q be formed from two copies of Z, identifying two fibers by a 90 degree rotation ψ(x1, x2) = (−x2, x1) extended to an orientation-reversing diffeomorphism of their normal bundles. We can do this so sections of the two copies of Z fit together to give a genus 2 surface with self-intersection 0 in Q.

Since the surfaces F of genus 2 in P and Q both have Euler number 0, there is an orientation-reversing diffeomorphism of their normal bundles. We can fix an identification of the normal bundle of F in P with F ×D2, take k copies of F ×p for k different points p close enough to 0 ∈ D2 to make F × p symplectic. We 2 can then form the fiber sum P #kQ. Now since (c1, c2)(Q) = (0, 0) and we are summing along a surface of genus 2, Proposition 4.4 tells us that summing with 2 k copies of Q increases (c1, c2) by (8k, 4k). Thus we have realized the invariants (8k − 3, 4k + 15) for all k ≥ 1, and it is easy to check that all pairs satisfying the constraints in the theorem statement can be obtained from these by blowing up. (See Proposition 4.1.)

We still need to see that P #kQ is simply connected. This is achieved by showing that any loop in a copy of Q \ (F × D2), where F is the surface obtained by piecing together the two sections, is trivial in π1(Q \ F ). We omit the details of this here. Then note that π1(P \ kF ) is generated by k meridians to the copies of F which were removed. These are homotopic to loops in the copies of Q and so to constant loops.

References

[BPVdV84] W. Barth, C. Peters, and A. Van de Ven. Compact complex surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1984. [CdS01] Ana Cannas da Silva. Lectures on symplectic geometry, volume 1764 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2001. [Don96] S. K. Donaldson. Symplectic submanifolds and almost-complex geometry. J. Differential Geom., 44(4):666–705, 1996. [Don98] S. K. Donaldson. Lefschetz fibrations in symplectic geometry. In Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), pages 309–314 (electronic), 1998. [GH94] Phillip Griffiths and Joseph Harris. Principles of algebraic geometry. Wiley Classics Library. John Wiley & Sons Inc., New York, 1994. Reprint of the 1978 original.

24 [Gom95] Robert E. Gompf. A new construction of symplectic manifolds. Ann. of Math. (2), 142(3):527–595, 1995. [GS99] Robert E. Gompf and AndrásI. Stipsicz. 4-manifolds and Kirby calculus, volume 20 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1999. [Liu96] Ai-Ko Liu. Some new applications of general wall crossing formula, Gompf’s conjecture and its applications. Math. Res. Lett., 3(5):569–585, 1996. [Sti00] A. I. Stipsicz. The geography problem of 4-manifolds with various structures. Acta Math. Hungar., 87(4):267–278, 2000. [Sti02] AndrásI. Stipsicz. Singular fibers in Lefschetz fibrations on mani- + folds with b2 = 1. Topology Appl., 117(1):9–21, 2002. [Tau96] Clifford H. Taubes. SW ⇒ Gr: from the Seiberg-Witten equations to pseudo-holomorphic curves. J. Amer. Math. Soc., 9(3):845–918, 1996. [Thu76] W. P. Thurston. Some simple examples of symplectic manifolds. Proc. Amer. Math. Soc., 55(2):467–468, 1976.